Handling the measurement error problem by means of panel data: Moment methods applied on firm data

etc. Here X _i(_tθ) denotes the [(T - 2) × K] matrix of x levels obtained by deleting
rows t and θ from X_i., and ∆X_i(t) denotes the [(T - 2) × K] matrix of x differences
obtained by stacking all one-period differences between rows of X_i. not including
period t and the single two-period difference between the columns for periods t +1
and t — 1. The vectors yi(_tθ) and ∆yi(_t) are constructed from y_i∙ in a similar way.
In general, we let subscripts (tθ) and (t) on a matrix or vector denote deletion of
(tθ) differences and t levels, respectively. Stacking y_i⁰_(tθ) , ∆y_i⁰_(t) , x_i(_tθ) , and ∆x_i(_t)
by individuals, we get

which have dimensions (N × (T — 2)), (N × (T — 2)), (N × (T — 2)K), and (N ×
(T — 2)K), respectively. These four matrices contain the alternative IV sets in the
GMM procedures to be considered below.

Equation in differences, IV’s in levels. Using X (_tθ) as IV matrix for ∆Xtθ ,we
obtain the following estimator of β, specific to period (t, θ) differences and utilizing
all admissible x level IV’s,

^hPi ^(∆xitθ^{) x}i(tθ) ^ihPi ^xi(tθ) ^xi(tθ) ^{i   hP}i ^xi

It exists if X ₍⁰_tθ)X _(tθ) has rank (T — 2)K, which requires N ≥ (T — 2)K. This
estimator examplifies (23), utilizes the OC E[x0(_tθ)(∆e_itθ)] = 0(T_-2)K, ₁ - which
follows from (32) - and minimizes the quadratic form

^[N-1X(⁰tθ)^∆tθ^]0[N-2X(⁰tθ)^X(tθ)^]-1[N-1X(⁰tθ)^∆tθ^].

18

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